Heat Flow

MME 528

 

1.  Physical Basics

            heat flow is due to three effects: 

            What is temperature?

                        see Java applet at:  http://mc2.cchem.berkeley.edu/Java/molecules/

                        a measure of the average kinetic energy of the molecules  ( mc2/2)

                                    so proportional to the mass of molecules and the square of the

                                    average speed

                        3 temperature scales:  Celsius, Fahrenheit,  Kelvin (absolute)

 

            What is heat?

                        the transfer of energy from one object to another?

 

            Experimental Fact #1

                        the heat energy Q of a body is proportional to its mass and to its absolute temperature T

 

                                    Q = cmT

                                    where  c = specific heat

 

            Experimental Fact #2

                        the rate of flow of heat energy through a unit area of surface is proportional to the temperature

                        difference on either side of the surface.  The heat flows in the direction of decreasing temperature

                        from hot to cold.

 

                        the constant of proportionality k is called the thermal conductivity and depends on the thickness of

the wall and the material

 

                        the units of  k are, in metric,   calories per degree Kelvin per cm2

 

                        This is sometimes referred to as Newton's Law of Cooling

 

2.  Mathematical relationships

 

            putting these together one has, after Δt units of time have elapsed,

 

                        (Q(t + Δt) – Q(t) )   =  - kA( T – Ts) Δt

 

            in the limit as  Δt -> 0 we get a derivative

 

                        dQ/dt =  -kA( T – Ts)

 

            but Q = cmT so   dQ/dt = cm dT/dt  so

 

                        cm dT/dt= -kA(T – Ts)

 

            or

                               dT/dt = - (kA/cm) (T- Ts)

 

          where

                                    T = temperature in degrees Kelvin

                                    t = time

                                    k = constant of thermal conductivity

                                    A = cross sectional area

                                    m = mass

                                    c = specific heat of the material

                                    Ts = temperature of surroundings (fixed)

 

3.   practical implementation

            the model is linear

            work in degrees Fahrenheit, ft, hours, BTUs

            compute heat flow per unit area

           

            result:   dQ/dt = - U A( T – Ts)    is the heat flow through a wall    (note changes)

                        where U depends on physical material and its thickness (replacing k)

                                    and is also per unit cross sectional area

                        and A is the cross sectional area of the object.

 

            further, the inverse of  U is called R and depends on the material only, including its thickness

                        that is,  U = 1/R

 

            Some sample U values:

 

                        door glass:   .650

                        roof insulation  (R – 19)   .049

                        wall insulation  (R – 11, wooden studs)   .088     3 1/2" standard walls

                        wooden door (solid)   .330

 

                        reference:  http://www.ci.kent.wa.us/PermitCenter/BuildingServices/HeatLossCalc.pdf

 

 

4. Sample House:

 

            single story ranch

            2 wooden doors

120 ft2 of windows -  single pane      

            8' walls           -   4" thick, insulated, wood studs

            45 degree roof  -  insulated with 6" of insulation

            40' x 24'  overall size

            outside temp: 20 degrees F

            inside :  70 degrees

 

            heat losses:

 

                        roof:

 

                        walls:

 

                        doors:

 

                        windows:

 

            total:                   (this is BTUs per hour).  Note:  1 gallon of heating oil has 139,000 BTU in it

            question: how many gallons of heating oil per day does this translate into??

 

Homework:

            semester schedule needed – last class is April 20th (14 meetings needed)

 

            1. for a wall which is 8' high and 20' long, with conventional studs (16" on center), what

                        % of area is insulation, what %  is wood?

 

2. spreadsheet

            total heat flow for a rectangular house given

                        wall height, length, width, window area, # doors, roof pitch

            columns for outside temperature: 30,20,10,0 degrees F

            standard house

            pie chart showing, for one temp, the comparative heat loss due to

                        walls, windows, doors, roof

            when done, compute the daily oil consumption

 

3.  more R and U values

            different windows

            floors

 

4.  heat loss of a house

            shut off heat, wait one hour

 

            5.  what is a calorie?? how is it related to a BTU?  what is a watt??

 

            6.  relationships of Celsius, Fahrenheit and Kelvin  scales

 

            7.  compare heat capacity and conductivity for the following:

 

                        wood (pine)

                        aluminum

                        Styrofoam

                        fiberglass insulation

                        concrete

                        brick

                        rock  (your choice)

                        water

                        air

           

            8.  volunteers needed to demo TI CBL temperature probe in 2 weeks (1/20)

                        Florence and Mark have volunteered…